Energy spaces for fractional Laplacians in bounded domains and homogeneous Dirichlet condition
Lately, I have discussed several times with colleagues about the “proper” functional set-up for the fractional Laplacian, in terms of energy spaces. We will focus on the case of Dirichlet homogeneous boundary/exterior conditions, and the three main operators in a smooth bounded domain $\Omega$:
\[\begin{aligned} (-\Delta)_{RFL}^s u (x) &= C_{d,s} PV \int_{\mathbb R^d} \frac{u(x) - u(y)}{|x-y|^{d+2s}} dy, \\ (-\Delta)_{CFL}^s u (x) &= C_{d,s} PV \int_{\Omega} \frac{u(x) - u(y)}{|x-y|^{d+2s}} dy, \end{aligned}\]and the spectral Dirichlet fractional Laplacian. If $(\varphi_i, \lambda_i)$ are the eigenvalues and $L^2$-orthonormal eigenfunctions of the Dirichlet Laplacian we define
\[(-\Delta_D)_{SFL}^s u (x) = \sum_{i=1}^\infty \lambda_i^s \langle u, \varphi_i \rangle_{L^2} \varphi_i (x).\]This discussion appeared already in [Bonforte, Sire & Vázquez, 2015], and the conclusion is that
\[\tag{S} \label{eq:interpolation L2 and H10} X_s = \begin{cases} H_0^s (\Omega) & \text{if } s \in (\frac 1 2, 1), \\ H_{00}^{\frac 1 2} (\Omega) & \text{if } s = \frac 1 2, \\ H^s (\Omega) & \text{if } s \in (0,\frac 1 2). \\ \end{cases}\]The definitions of these spaces are given below. However, there are some points in that paper which can be quite technical and we will try here to make a more introductory presentation.
Notation. Through this note we will use the notation
\[\delta(x) = \mathrm{dist}(x,\mathbb R^d \setminus \Omega)\]and \(f \asymp g\) will denote that there exists a constant $C = C(s,d, \Omega) > 0$ such that $C^{-1} g \le f \le C g$.
Bilinear forms
We can define the bilinear form (Dirichlet energy) associated to each as
\[\mathcal E_{\mathcal L} (u,v) = \int_\Omega u \mathcal L v\]for smooth functions, and extended in a suitable way.
RFL and SFL
These are related the Sobolev-Slobodeckii seminorm
\[[u]_{H^s (\Omega)} = \left( \int_\Omega \int_\Omega \frac{|u(x) - u(y)|^2}{|x-y|^{d+2s}} dx dy \right)^{\frac 1 2} .\]With this definition
\[\mathcal E_{CFL} (u,u) = C [u]_{H^s (\Omega)}^2, \qquad \mathcal E_{RFL} (u,u) = C [u]_{H^s (\mathbb R^d)}^2 .\]There is a well-known relationship between these operators. If $u = 0$ in $\mathbb R^d \setminus \Omega$ we can write
\[(-\Delta)^s_{RFL} u (x) = (-\Delta)_{CFL} u (x) + u(x) \kappa_{RFL}(x),\]where
\[\kappa_{RFL} (x) = C_{s,d} \int_{\mathbb R^d \setminus \Omega} \frac{dy}{|x-y|^{d+2s}}.\]Therefore we have the relation between the energies
\[\mathcal E_{RFL} (u,u) = C [u]_{H^s (\Omega)}^2 + \int_\Omega \kappa_{RFL}(x) u(x)^2 dx, \qquad \text{if } u = 0 \text{ in } \mathbb R^d \setminus \Omega.\]If $\Omega$ is Lipschitz then
\[\kappa_{RFL} (x) \asymp \delta(x)^{-2s}\]Eventually, we get the equivalence
\[\mathcal E_{RFL} (u,u) \asymp \| u\|^2_{H^s \cap L^2 (\delta^{-2s})} := [u]_{H^s}^2 + \int_\Omega \frac{u(x)^2}{\delta(x)^{2s}}, \qquad \forall u \in H^s \cap C_c (\Omega).\]This norm will appear again later.
SFL
For the SFL we get
\[\mathcal E_{SFL} (u,u) = \sum_i \lambda_i^s \langle u , \varphi_i \rangle^2.\]We will show below, through interpolation, that we also have \(\mathcal E_{SFL}(u,u) \asymp \| u\|^2_{H^s \cap L^2 (\delta^{-2s})}.\)
Let us give a succinct, informal, justification of this fact using a different representation of the operator.
First, we observe \(\lambda^s = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{-\lambda t} - 1) \frac{dt}{t^{1+s}}.\) and hence
\[(-\Delta_{D})^s_{SFL} u (x)= -\int_0^\infty\Big( e^{-t (-\Delta_{D})} [u](x) - u(x) \Big) d\mu(t).\]where $d \mu(t) = \frac{s}{\Gamma(1-s)} \frac{1}{t^{1+s}} \ge 0$.
Here $w = e^{-t (-\Delta_D)} f$ is the solution to
\[\begin{cases} \frac{\partial w}{\partial t} - \Delta w = 0 , & \text{in } (0,\infty) \times \Omega, \\ w(0,x) = f & \text{on } \{0\} \times \Omega \\ w = 0 & \text{on } (0,\infty) \times \partial \Omega. \end{cases}\]We can use the heat kernel $K_D(t,x,y) = e^{-t (-\Delta_{D})} [\delta_y] (x)$ to show that
\[e^{-t (-\Delta_{D})} [u](x) = \int_\Omega K(t,x,y) u(y) dy\]Since $S_{D}$ is self-adjoint, then $K_D(t,x,y) = K_D(t,y,x) \ge 0$. We can compute
\[\begin{aligned} (-\Delta_{D})^s_{SFL} u (x)&= - \int_0^\infty \left(\int_\Omega K_D(t,x,y) u(y) dy - u(x) \right) d\mu(t) \\ &= - \int_0^\infty \int_\Omega K_D(t,x,y) (u(y) - u(x)) dy d\mu(t) \\ &\qquad + u(x) \int_0^\infty \left(1 - \int_\Omega K_D(t,x,y) dy \right)d\mu_t \end{aligned}\]We recover the jumping kernel
\[J_{SFL}(x,y) = \int_0^\infty K_D(t,x,y) d \mu(t) \ge 0\]and the killing kernel
\[\kappa_{SFL} (x) = \int_0^\infty \left(1 - \int_\Omega K_D(t,x,y) dy \right)d\mu_t\]Notice that the Dirichlet boundary conditions imply loss of mass with time, and convergence of the mass to $0$.
The energy is therefore
\[\mathcal E_{SFL}(u,u) = \int_\Omega \int_\Omega J(x,y) (u(x) - u(y))^2 dx dy + \int_\Omega \kappa_{SFL} (x) u(x)^2 dx.\]much like in the previous cases.
It was proven by [Song & Vondraček, 2003] show by probabilistic arguments that
\[\kappa_{SFL}(x) \asymp \delta(x)^{-2s}\]Energy spaces
The fractional Hardy inequality
Let us consider the fractional Hardy inequality
\[[u]_{H^s(\Omega)}^2 \ge \mathsf H_{d,s} \int_\Omega \frac{u(x)^2}{\delta(x)^{2s}} \qquad \forall u \in C_c (\Omega).\]where $\mathsf H_{d,s}$ is the optimal constant. We only care about its positivity.
If $\Omega$ is $C^{1,1}$ we can locally remap its boundary to a piece of $[0,\infty) \times \mathbb R^{d-1}$. In this setting, it was proven by [Bogdan & Dyda, 2011] that
\[\mathsf H_{d,s} = \begin{cases} > 0 & \text{if } s \ne \frac 1 2, \\ = 0 & \text{if } s = \frac 1 2. \end{cases}\]The case $s = \frac 1 2$ is easy to check. The function $u(x) = 1$ can be approximated in $H^{\frac 1 2}$ by a sequence $u_k \in H^{\frac 1 2} (\Omega) \cap C_c(\Omega)$. Using the tubular neighbourhood theorem $|u_k / \sqrt{d}|_{L^2} \to \infty$. Hence, we have the equivalence of norms
\[[u]_{H^s} \le \| u \|_{H^s \cap L^2(\delta^{-2s})} \le \sqrt{1 + \mathsf H_{d,s}^{-1}} [u]_{H^s} \qquad \forall s \ne \frac 1 2.\]and the equivalence fails for $s = \frac 1 2$.
Sobolev spaces
We define the $H^s$ norm as
\[\|u\|_{H^s} = [u]_{H^s} + \| u \|_{L^2}\]and the space
\[H^s (\Omega) = \overline{C^\infty (\Omega)}^{H^s}.\]To add the boundary condition, we define the homogeneous version
\[H^s_0 (\Omega) = \overline{C_c^\infty (\Omega)}^{H^s}.\]It turns out that
\[H^s_0 (\Omega) \begin{cases} = H^s (\Omega), &\text{if } s \in (0,\tfrac 1 2], \\ \ne H^s (\Omega), &\text{if } s \in (\tfrac 1 2,1]. \end{cases}\]So $H^s_0(\Omega)$ is the natural setting for RFL and SFL for $s \ne \frac 1 2$. In order to have pointwise boundary conditions, the CFL is only defined for $s > \frac 1 2$.
But the killing kernel $\kappa_{SFL} , \kappa_{RFL} \asymp \delta^{-2s}$. So $H^{\frac 1 2}_0(\Omega)$ is not the correct setting for $s = \frac 1 2$.
The natural setting for these problems is, in fact, the Lions-Magenes space
\[H^{\frac 1 2}_{00} (\Omega) = \{ u \in H^{\frac 1 2} (\Omega) : \tfrac{u}{\sqrt{\delta}} \in L^2 (\Omega)\}\]See [Tartar, 2007] Chapter 33. Because of energy characterisation above, this is the good space for the RFL when $s = \frac 1 2$.
Again $u(x) = 1$ is clearly \(u \notin H^{\frac 1 2}_{00} (\Omega)\). Hence
\[H^{\frac 1 2}_{00} (\Omega) \subsetneq H^{\frac 1 2}_{0} (\Omega) = H^{\frac 1 2} (\Omega).\]Furthermore, this is a short proof that $\mathsf H_{d,\frac 1 2} = 0$.
Interpolation theory
Definition
Provided spaces \(X_0, X_1\) we define the real interpolation by the \(K\)-method as follows. For \(u \in X_0 + X_1\)
\[K(t,u;X_0,X_1) = \inf_{u = u_0 + u_1} (\| u_0 \|_{X_0} + t \|u_1\|_{X_1}).\]The interpolation norm is defined for $s \in (0,1)$ and $q \in [1,\infty)$
\[\| u \|_{\theta,q;K} = \left( \int_0^\infty t^{-q\theta} K(t,u; X_0,X_1)^q \frac{dt}{t} \right)^{\frac 1 q}.\]We define \([X_0, X_1]_{\theta,q}\) as the set of elements with finite norm.
A classical result is that
\[[L^2(\Omega), H^1(\Omega)]_{s,2;K} = H^s (\Omega).\]It can be found in [Tartar, 2007] Chapters 34-36 that with definition
\[[L^2(\Omega), H_0^1(\Omega)]_{\theta,2;K} = X_s\]The Lions-Magenes spaces and Hardy’s inequality
Notice that there is also a Hardy inequality for $s = 1$ and for $L^2$ we have the trivial \(\int_\Omega u^2 / \delta^0 = \| u \|_{L^2}^2\). Hence, by interpolation
\[\| u \|_{\theta, 2; K}^2 \ge C_s {\mathsf H}_{d,1}^s \int_\Omega \frac{u^2}{\delta^{2s}} .\]where ${\mathsf H}_{d,1} > 0$. Therefore, the interpolation space for \(s = 1\) is not \(X_{\frac 1 2} = H_{00}^{\frac 1 2} (\Omega)\) and not \(H_0^s (\Omega)\). Another way of reading this result is that
\[X_s = [L^2(\Omega), H^1(\Omega) \cap L^2 (\delta^{-2})]_{\theta,2} = H^s (\Omega) \cap L^2 (\delta^{-2s}).\]It is therefore not hard to see that
\[\| u \|_{s,2;K} \asymp \| u \|_{H^s \cap L^2(\delta^{-2s})}.\]The norm in the sense of weights is “more stable” in the interpolation that the density of $C_c^\infty$ functions.
The eigen-decomposition of the Dirichlet Laplacian \(s=1\)
It is a direct computation that if $u \in H_0^1 (\Omega)$
\[[u]_{H^1} = \int_\Omega |\nabla u|^2 = \int_\Omega u (-\Delta u) = \sum_{i=1} \lambda_i \langle u , \varphi_i \rangle^2.\]For Hilbert spaces it is more natural to use
\[\overline K(t,u;X_0,X_1) = \inf_{u = u_0 + u_1} \sqrt{\| u_0 \|_{X_0}^2 + t^2 \|u_1\|_{X_1}^2}.\]The norm generated is equivalent to that with $K$ since
\[\frac 1 2 \left( |a| + |b|\right) \le \sqrt{ |a|^2 + |b|^2} \le |a| + |b|\]Let $u(x) = \sum \langle u, \varphi_i\rangle_{L^2} \varphi_i(x)$ and $u_0 = \sum b_i \varphi_i$ and we can compute the \(\| u_0 \|_{L^2}^2 + t^2 \|u - u_0\|_{H^1_0}^2\) by expanding in the eigenbasis. To compute the minimum on $u_0$ take a derivative w.r.t. each \(b_i\) to deduce \(b_i = \frac{1}{1 + t^2 \lambda_i^2}\). Eventually,
\[\overline K(t,u; L^2, H_0^1)^2 = \sum_i \frac{t^2 \lambda_i^2}{1 + t^2 \lambda_i^2 } \langle u, \varphi_i\rangle_{L^2}^2.\]Hence
\[\int_0^\infty t^{-2\theta} \overline K(t,u; X_0,X_1)^2 \frac{dt}{t} = C_s \sum_{i} \lambda_i^s \langle u, \varphi_i\rangle^2.\]This is precisely the energy norm for the SFL. We summarise the results as
Theorem. For all \(s \in (0,1)\) and \(u \in H^s (\Omega) \cap C_c(\Omega)\), then
\[\tag{E} \label{eq:equivalence} \mathcal E_{SFL} (u,u) \asymp ||u ||_{s,2;K} \asymp [u]_{H^s}^2 + \int_\Omega \frac{u(x)^2}{\delta(x)^{-2s}} \asymp \mathcal E_{RFL} (u,u).\]The correct space for homogeneous Dirichlet conditions is $X_s$
Remark. The case $s = \frac 1 2$ is delicate and has caused some slight errors in the literature. Relation \eqref{eq:equivalence} was also proved in Lemma 4.2 in [Song & Vondraček, 2003]. However, their proof seems to be false for $s = \frac 1 2$. The authors cite [Chen & Song, 2003] . where the authors prove the complex interpolation
\[[L^2(\Omega), H^1_0(\Omega)]_s = H^s_0 (\Omega), \qquad \forall s \in (0,\tfrac 1 2) \cup (\tfrac 1 2,1),\]and in section 3 they assumme that it holds also for $s = \frac 1 2$. In [Song & Vondraček, 2003], authors use $H_0^s (\Omega)$ also for $s = \frac 1 2$.
References
- Bonforte, Matteo and Sire, Yannick and Vázquez, Juan Luis (2015) . Existence, Uniqueness and Asymptotic Behaviour for Fractional Porous Medium Equations on Bounded Domains. Discret. Contin. Dyn. Syst. Ser. A. Link
- Song, Renming and Vondraček, Zoran (2003) . Potential Theory of Subordinate Killed Brownian Motion in a Domain. Probability Theory and Related Fields. Link
- Bogdan, Krzysztof and Dyda, Bartłomiej (2011) . The Best Constant in a Fractional Hardy Inequality. Mathematische Nachrichten. Link
- Tartar, Luc (2007) . An Introduction to Sobolev Spaces and Interpolation Spaces. Springer Berlin Heidelberg. Link
- Chen, Zhen-Qing and Song, Renming (2003) . Hardy Inequality for Censored Stable Processes. Tohoku Mathematical Journal. Link
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